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Abstract
This paper presents a graphical solution to model fluid flow in permeable media in the presence of compressibility. Analytical solutions are important as numerical simulations do not yield explicit expressions in terms of the model parameters. Furthermore, simulations that provide the most comprehensive solutions to multiphase flow problems are computationally intensive.

The method of characteristics (MOC) solution of the overall mass conservation equation of CO2 in two-phase two-component flow through permeable media is derived while considering the compressibility of fluids and the rock. The previously developed MOC solutions rely on the incompressible fluid and rock assumptions that are rarely met in practice; hence, the incompressible assumption is relaxed and the first graphical/analytical solution for compressible flow is derived. The analytical solution is validated by simulation results.

The results suggest that the velocity of a wave, which is associated with the transport of a certain mass of CO2 along the permeable medium, is a function of compressibility of the rock and fluid, fractional flow terms, gas saturation, and the slope of fractional flow curve. Furthermore, the wave velocity will be only function of fractional flow terms, gas saturation, and the slope of fractional flow curve if the compressibility of the rock is negligible compared to that of CO2.

Thus, this paper explains how fast a compressible CO2 plume will travel along the aquifers length. In practice, the fate of the injected CO2 plume is essential to determine the storage capacity of aquifers and to evaluate the risk associated with the CO2 sequestration projects.

Introduction
Despite extensive research on analytical modeling of CO2 sequestration in saline aquifers (Szulczewski et al., 2009; Juanes et al., 2010; Ghanbarnezhad et al., 2011), the gas always has been considered as an incompressible fluid. The method of characteristics (MOC) solution of the overall composition balance equation of CO2 is derived for one-dimensional (1D) two-phase two-component flow in the presence of compressibility. In the following study, the incompressible assumption is relaxed as unequal injection and discharge rates occur more often in practice; the unequal rates exhibit the compressibility of fluids. Note that with zero compressibility involved, it is impossible to inject more than the discharge rate in an aquifer. Thus, the total flow velocity (gaseous +aqueous) stays constant with distance when compressibility is absent; on the contrary, it can vary in the presence of compressibility; the continuity equation necessitates this statement.